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A Favard theorem for orthogonal rational functions on the unit circle

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Abstract

We consider forn=0, 1,... the nested spaces ℒ n of rational functions of degreen at most with given poles\(1/\bar \alpha _i , |\alpha _i |< 1, i = 1,...,n\). Given a finite measure supported on the unit circle, we associate with it a nested orthogonal basis of rational functions Φ0,...,Φ n for ℒ n ,n=0, 1,.... These Φ n satisfy a recurrence relation that generalizes the recurrence for Szegő polynomials.

In this paper we shall prove a Favard type theorem which says that if one has a sequence of rational functions Φ n ∈ ℒ n which are generated by such a recurrence, then there will be a measure μ supported on the unit circle to which they are orthogonal. We shall give a sufficient condition for the uniqueness of this measure.

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Authors and Affiliations

  1. Department of Computer Science, K.U. Leuven, B-3001, Leuven, Belgium

    Adhemar Bultheel

  2. Facultad de Matemáticas, Universidad de La Laguna, La Laguna, Tenerife, Canary Islands, Spain

    Pablo González-Vera

  3. Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands

    Erik Hendriksen

  4. Department of Mathematics, University of Trondheim-NTH, N-7034, Trondheim, Norway

    Olav Njåstad

Authors
  1. Adhemar Bultheel
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  2. Pablo González-Vera
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  3. Erik Hendriksen
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  4. Olav Njåstad
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Bultheel, A., González-Vera, P., Hendriksen, E. et al. A Favard theorem for orthogonal rational functions on the unit circle. Numer Algor 3, 81–89 (1992). https://doi.org/10.1007/BF02141918

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